Algebraic metalanguages for routing policy specification Alexander - PowerPoint PPT Presentation

Algebraic metalanguages for routing policy specification Alexander Gurney Computer Laboratory, University of Cambridge Multi-Service Networks 2006 Cosener’s House, Abingdon, UK

The Metarouting Idea • A language for policy • Build up complex protocol descriptions, as simple algebras combined in known ways • Susceptible to proof • Suitable for implementation • One language, many meanings

Meanwhile, in the world of mathematics • The shortest path problem, with numbers (Dijkstra, Bellman / Ford, Moore / Shannon; and related works back to at least 1871) • Generalised shortest paths (Gondran / Minoux; Carré; Berge; ...) • All kinds of fun algebra: monoids, semirings, semilattices and other ordered structures, actions, representations • “If the algebra has property X then we can use algorithm Y to find an optimal solution” for various values of “X”, “Y”, and “optimal”

Model I: Path algebra • Monoid (M, +, 0): closed, associative, identity • Semiring (S, ∧ , ⊕ , ⊤ , 0): two monoids (S, ∧ , ⊤ ), (S, ⊕ , 0) on the same set • Path algebra: has commutativity, distributivity, idempotence ( a + a = a ) • Examples: ( N ∞ , min, +, ∞ , 0), ( ℘ (A), ∩ , ∪ , A, ∅ ), ( R + , max, min, 0, ∞ )

Model I: Path algebra • Monoid (M, +, 0): closed, associative, identity • Semiring (S, ∧ , ⊕ , ⊤ , 0): two monoids (S, ∧ , ⊤ ), (S, ⊕ , 0) on the same set • Path algebra: has commutativity, distributivity, idempotence ( a + a = a ) • Examples: ( N ∞ , min, +, ∞ , 0), ( ℘ (A), ∩ , ∪ , A, ∅ ), ( R + , max, min, 0, ∞ ) 2 6 4 4 0 2 5 1 9

Model I: Path algebra • Monoid (M, +, 0): closed, associative, identity • Semiring (S, ∧ , ⊕ , ⊤ , 0): two monoids (S, ∧ , ⊤ ), (S, ⊕ , 0) on the same set • Path algebra: has commutativity, distributivity, idempotence ( a + a = a ) • Examples: ( N ∞ , min, +, ∞ , 0), ( ℘ (A), ∩ , ∪ , A, ∅ ), ( R + , max, min, 0, ∞ ) 2 6 4 0+2+4+6+4 4 0 2 5 1 9

Model I: Path algebra • Monoid (M, +, 0): closed, associative, identity • Semiring (S, ∧ , ⊕ , ⊤ , 0): two monoids (S, ∧ , ⊤ ), (S, ⊕ , 0) on the same set • Path algebra: has commutativity, distributivity, idempotence ( a + a = a ) • Examples: ( N ∞ , min, +, ∞ , 0), ( ℘ (A), ∩ , ∪ , A, ∅ ), ( R + , max, min, 0, ∞ ) 2 6 4 16 4 0 2 5 1 9

Model I: Path algebra • Monoid (M, +, 0): closed, associative, identity • Semiring (S, ∧ , ⊕ , ⊤ , 0): two monoids (S, ∧ , ⊤ ), (S, ⊕ , 0) on the same set • Path algebra: has commutativity, distributivity, idempotence ( a + a = a ) • Examples: ( N ∞ , min, +, ∞ , 0), ( ℘ (A), ∩ , ∪ , A, ∅ ), ( R + , max, min, 0, ∞ ) 2 6 4 16 4 0 0+5+2+6+4 2 5 1 9

Model I: Path algebra • Monoid (M, +, 0): closed, associative, identity • Semiring (S, ∧ , ⊕ , ⊤ , 0): two monoids (S, ∧ , ⊤ ), (S, ⊕ , 0) on the same set • Path algebra: has commutativity, distributivity, idempotence ( a + a = a ) • Examples: ( N ∞ , min, +, ∞ , 0), ( ℘ (A), ∩ , ∪ , A, ∅ ), ( R + , max, min, 0, ∞ ) 2 6 4 16 4 0 17 2 5 1 9

Model I: Path algebra • Monoid (M, +, 0): closed, associative, identity • Semiring (S, ∧ , ⊕ , ⊤ , 0): two monoids (S, ∧ , ⊤ ), (S, ⊕ , 0) on the same set • Path algebra: has commutativity, distributivity, idempotence ( a + a = a ) • Examples: ( N ∞ , min, +, ∞ , 0), ( ℘ (A), ∩ , ∪ , A, ∅ ), ( R + , max, min, 0, ∞ ) 2 6 4 16 4 0 17 2 5 1 9 0+5+9+1

Model I: Path algebra • Monoid (M, +, 0): closed, associative, identity • Semiring (S, ∧ , ⊕ , ⊤ , 0): two monoids (S, ∧ , ⊤ ), (S, ⊕ , 0) on the same set • Path algebra: has commutativity, distributivity, idempotence ( a + a = a ) • Examples: ( N ∞ , min, +, ∞ , 0), ( ℘ (A), ∩ , ∪ , A, ∅ ), ( R + , max, min, 0, ∞ ) 2 6 4 16 4 0 17 2 5 1 9 15

Model I: Path algebra • Monoid (M, +, 0): closed, associative, identity • Semiring (S, ∧ , ⊕ , ⊤ , 0): two monoids (S, ∧ , ⊤ ), (S, ⊕ , 0) on the same set • Path algebra: has commutativity, distributivity, idempotence ( a + a = a ) • Examples: ( N ∞ , min, +, ∞ , 0), ( ℘ (A), ∩ , ∪ , A, ∅ ), ( R + , max, min, 0, ∞ ) 2 6 4 16 4 0 17 2 5 1 9 15

Model I: Path algebra • Monoid (M, +, 0): closed, associative, identity • Semiring (S, ∧ , ⊕ , ⊤ , 0): two monoids (S, ∧ , ⊤ ), (S, ⊕ , 0) on the same set • Path algebra: has commutativity, distributivity, idempotence ( a + a = a ) • Examples: ( N ∞ , min, +, ∞ , 0), ( ℘ (A), ∩ , ∪ , A, ∅ ), ( R + , max, min, 0, ∞ )

Model I: Path algebra • Monoid (M, +, 0): closed, associative, identity • Semiring (S, ∧ , ⊕ , ⊤ , 0): two monoids (S, ∧ , ⊤ ), (S, ⊕ , 0) on the same set • Path algebra: has commutativity, distributivity, idempotence ( a + a = a ) • Examples: ( N ∞ , min, +, ∞ , 0), ( ℘ (A), ∩ , ∪ , A, ∅ ), ( R + , max, min, 0, ∞ ) 2 5.8 4.5 4 ∞ 2.3 5 1 9.1

Model I: Path algebra • Monoid (M, +, 0): closed, associative, identity • Semiring (S, ∧ , ⊕ , ⊤ , 0): two monoids (S, ∧ , ⊤ ), (S, ⊕ , 0) on the same set • Path algebra: has commutativity, distributivity, idempotence ( a + a = a ) • Examples: ( N ∞ , min, +, ∞ , 0), ( ℘ (A), ∩ , ∪ , A, ∅ ), ( R + , max, min, 0, ∞ ) 2 5.8 4.5 {2, 4, 5.8, 4.5} 4 ∞ 2.3 5 1 9.1

Model I: Path algebra • Monoid (M, +, 0): closed, associative, identity • Semiring (S, ∧ , ⊕ , ⊤ , 0): two monoids (S, ∧ , ⊤ ), (S, ⊕ , 0) on the same set • Path algebra: has commutativity, distributivity, idempotence ( a + a = a ) • Examples: ( N ∞ , min, +, ∞ , 0), ( ℘ (A), ∩ , ∪ , A, ∅ ), ( R + , max, min, 0, ∞ ) 2 5.8 4.5 2 4 ∞ 2.3 5 1 9.1

Model I: Path algebra • Monoid (M, +, 0): closed, associative, identity • Semiring (S, ∧ , ⊕ , ⊤ , 0): two monoids (S, ∧ , ⊤ ), (S, ⊕ , 0) on the same set • Path algebra: has commutativity, distributivity, idempotence ( a + a = a ) • Examples: ( N ∞ , min, +, ∞ , 0), ( ℘ (A), ∩ , ∪ , A, ∅ ), ( R + , max, min, 0, ∞ ) 2 5.8 4.5 2 4 ∞ {5, 2.3, 5.8, 4.5} 2.3 5 1 9.1

Model I: Path algebra • Monoid (M, +, 0): closed, associative, identity • Semiring (S, ∧ , ⊕ , ⊤ , 0): two monoids (S, ∧ , ⊤ ), (S, ⊕ , 0) on the same set • Path algebra: has commutativity, distributivity, idempotence ( a + a = a ) • Examples: ( N ∞ , min, +, ∞ , 0), ( ℘ (A), ∩ , ∪ , A, ∅ ), ( R + , max, min, 0, ∞ ) 2 5.8 4.5 2 4 ∞ 2.3 2.3 5 1 9.1

Model I: Path algebra • Monoid (M, +, 0): closed, associative, identity • Semiring (S, ∧ , ⊕ , ⊤ , 0): two monoids (S, ∧ , ⊤ ), (S, ⊕ , 0) on the same set • Path algebra: has commutativity, distributivity, idempotence ( a + a = a ) • Examples: ( N ∞ , min, +, ∞ , 0), ( ℘ (A), ∩ , ∪ , A, ∅ ), ( R + , max, min, 0, ∞ ) 2 5.8 4.5 2 4 ∞ 2.3 2.3 5 1 9.1 {5, 9.1, 1}

Model I: Path algebra • Monoid (M, +, 0): closed, associative, identity • Semiring (S, ∧ , ⊕ , ⊤ , 0): two monoids (S, ∧ , ⊤ ), (S, ⊕ , 0) on the same set • Path algebra: has commutativity, distributivity, idempotence ( a + a = a ) • Examples: ( N ∞ , min, +, ∞ , 0), ( ℘ (A), ∩ , ∪ , A, ∅ ), ( R + , max, min, 0, ∞ ) 2 5.8 4.5 2 4 ∞ 2.3 2.3 5 1 9.1 1

Model I: Path algebra • Monoid (M, +, 0): closed, associative, identity • Semiring (S, ∧ , ⊕ , ⊤ , 0): two monoids (S, ∧ , ⊤ ), (S, ⊕ , 0) on the same set • Path algebra: has commutativity, distributivity, idempotence ( a + a = a ) • Examples: ( N ∞ , min, +, ∞ , 0), ( ℘ (A), ∩ , ∪ , A, ∅ ), ( R + , max, min, 0, ∞ ) 2 5.8 4.5 2 4 ∞ 2.3 2.3 5 1 9.1 1

Combining operations • D = ( N ∞ , min, +, ∞ , 0), B = ( R + , max, min, 0, ∞ ) • D × B = ( N ∞ × R + , ∧ , ⊕ , ( ∞ , 0), (0, ∞ )), where ( d , b ) ∧ ( e , c ) = (min( d , e ), max( b , c )) ( d , b ) ⊕ ( e , c ) = ( d + e , min( b , c )) • D × lex B = ( N ∞ × R + , ∧ lex , ⊕ , ( ∞ , 0), (0, ∞ )), where ( d , b ) ∧ lex ( e , c ) = ( d , b ) if d < e or [ d = e and b > c ]; ( b , c ) otherwise ⊕ is as before We obtain < from min by defining a ≤ b iff a = min( a , b )

Dijkstra’s algorithm and the prefix property • To get the right answer out of Dijkstra, we need each prefix of a shortest path to also be a shortest path • In semiring language: a = a ∧ b ⇒ ( m + a ) = ( m + a ) ∧ ( m + b ), for all a , b , m a m b • This is the case when we have distributivity: ( m + a ) ∧ ( m + b ) = m + ( a ∧ b )

Property preservation • If A and B are distributive, then so is A × B; but A × lex B may not be • So if we have a whole load of distributive semirings A, B, C, and D, then we know we can run Dijkstra correctly on A × B × C × D • Similar rules exist for other properties and other operations – so we can deduce facts like “we can’t do Dijkstra, but we can do Bellman-Ford”